FLOWOOD, Miss. — Senator Thad Cochran’s supporters opened Mississippi’s Republican Senate runoff on Wednesday by signaling that they would treat the race like a general election and seek the votes of Democrats and independents during the three-week campaign against State Senator Chris McDaniel.

“I don’t care if you’re from Yazoo City or the Coast, or if you are white or black,” said Henry Barbour, a Republican activist who runs Mississippi Conservatives, a pro-Cochran political action committee. “We’re going to make certain that everybody knows they have a stake in this, and the state needs somebody to represent the interest of all three million Mississippians.”

The runoff election is June 24, and “every registered voter in Mississippi has a stake in this, and needs to show up,” he added.

In separate comments, Austin Barbour, a Cochran campaign aide and Henry Barbour’s brother, was not as explicit, but he also made it clear that the senator would widen his search for voters. “If you live in Mississippi and you can vote, you have the opportunity to cast a vote in this election,” he said. (The brothers are, by law, not allowed to communicate about the campaign.)

It was not officially clear until Wednesday afternoon that a runoff would be needed. For most of the vote count on Tuesday night, both candidates hovered around 49 percent, with the potential that one of them could avoid the runoff by pushing past 50 percent. But that did not happen, and when the totals were announced, Mr. Cochran was behind Mr. McDaniel, his Tea Party-backed rival, by 1,386 votes.

Henry Barbour’s comments were an indication that Mr. Cochran’s campaign supporters recognized the urgency in devising a way to counter the ardent Tea Party supporters who are likely to return to the polls on Mr. McDaniel’s behalf.

Mr. McDaniel’s supporters ridiculed the possibility that Mr. Cochran could draw significant numbers of Democrats and independents to defeat the challenger.

“The Cochran campaign is by all means welcome to try to make this runoff completely different from every other runoff that has been run in the history of American politics,” said Barney Keller, a spokesman for the conservative Club for Growth, which spent the most of the conservative groups against Mr. Cochran during the primary.

There is no party registration in Mississippi, and with former Representative Travis Childers having secured the Democratic Senate nomination on Tuesday, there will be no statewide Democratic runoff.

Still, luring Democrats and independents to cast ballots in an early-summer Republican runoff is sure to be difficult, with such contests typically drawing fewer voters than the primaries. The effort to do so shows the straits that Mr. Cochran, 76, suddenly finds himself in after four decades representing Mississippi in Congress. He may have no other way to get past Mr. McDaniel, 41, than to broaden the electorate.

As conservative as Mississippi is — and that was illustrated by Mr. McDaniel’s strong showing — the state also has the highest percentage of black residents: 37 percent. African-Americans are an overwhelmingly Democratic constituency, but Mr. Cochran’s supporters believe they could be persuaded to turn out for the senator to stop Mr. McDaniel because of some racially provocative comments he made as a conservative talk radio host.

But the Cochran campaign faces a significant complication: state law bars voters who cast ballots in one party’s primary from participating in another party’s runoff. “Good luck with that,” Mr. Keller said. “In runoffs, turnout goes down and more motivated voters show up.”

A spokesman for Mr. McDaniel himself did not comment, and the candidate himself appeared to stay out of sight on Wednesday.

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Jonathan Martin on the race in Mississippi as G.O.P. Senate candidates prepare for a runoff.Published OnJune 4, 2014CreditImage by George Clark/Associated Press

Mr. Cochran made only a brief public appearance, strolling with a local congressman into a Jackson-area Chick-fil-A late in the afternoon to a standing ovation. The senator, who did not speak on election night, vowed to work hard in the runoff but offered little clue about his plans. “I want to be surprised,” he said. “That way I’m always fresh.”

Both candidates must move to replenish their campaign finances for the next phase of the race, which under campaign finance rules is considered a separate election. That will allow the candidates to return to contributors who previously gave the maximum allowed.

But the bigger question looming is not what Mr. Cochran and Mr. McDaniel can raise, but how much support they will get from the outside groups, which spent a total of more than $8 million in the primary.

Henry Barbour said Mississippi Conservatives had spent most of its money in the run-up to the primary but that it would move to “reload” financially to get commercials back on television.

Officials with the National Republican Senatorial Committee said the party’s leadership remained committed to Mr. Cochran, and there were discussions on Wednesday about how to help him raise money for the runoff.

Less clear were the intentions of the U.S. Chamber of Commerce, which spent about $500,000 on Mr. Cochran during the primary. Chamber officials said Wednesday they continued to back the senator, but they were deliberating on whether that meant spending additional money on his behalf.

The Club for Growth, which has already put in $2.5 million on Mr. McDaniel’s campaign, released a statement promising to “vigorously pursue this race to its conclusion.”

Nick Corasaniti reported from Flowood, and Jonathan Martin from Washington.

A version of this article appears in print on , on Page A16 of the New York edition with the headline: Incumbent in Mississippi Runoff Aims for More Than Just G.O.P. Votes. Order Reprints | Today’s Paper | Subscribe